Kähler differential

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

Definition

Let $R$ and $S$ be commutative rings and $φ : R \to S$ be a ring homomorphism. An important example is for $R$ a field and $S$ a unital algebra over $R$ (such as the coordinate ring of an affine variety). Kähler differentials formalize the observation that the derivatives of polynomials are again polynomial. In this sense, differentiation is a notion which can be expressed in purely algebraic terms. This observation can be turned into a definition of the module

\Omega _{S/R}.

of differentials in different, but equivalent ways.

Definition using derivations

An $R$ -linear derivation on $S$ is an $R$ -module homomorphism $d\colon S\to M$ to an $S$ -module $M$ with the image of $R$ in its kernel, satisfying the Leibniz rule $d(fg)=f\,dg+g\,df$ . The module of Kähler differentials is defined as the $S$ -module $\Omega _{S/R}$ for which there is a universal derivation $d\colon S\to \Omega _{S/R}$ . As with other universal properties, this means that $d$ is the best possible derivation in the sense that any other derivation may be obtained from it by composition with an $S$ -module homomorphism. In other words, the composition with $d$ provides, for every $S -module$ $M$ , an $S$ -module isomorphism

\operatorname {Hom} _{S}(\Omega _{S/R},M){\xrightarrow {\cong }}\operatorname {Der} _{R}(S,M).

One construction of $Ω S / R$ and $d$ proceeds by constructing a free $S$ -module with one formal generator $ds$ for each $s$ in $S$ , and imposing the relations

$dr = 0$ ,
$d (s + t) = ds + dt$ ,
$d (st) = s dt + t ds$ ,

for all $r$ in $R$ and all $s$ and $t$ in $S$ . The universal derivation sends $s$ to $ds$ . The relations imply that the universal derivation is a homomorphism of $R$ -modules.

Definition using the augmentation ideal

Another construction proceeds by letting $I$ be the ideal in the tensor product $S\otimes _{R}S$ defined as the kernel of the multiplication map $S\otimes _{R}S\to S$ given by $\Sigma s_{i}\otimes t_{i}\mapsto \Sigma s_{i}\cdot t_{i}$ . Then the module of Kähler differentials of $S$ can be equivalently defined by^[1] $Ω S / R = I / I 2$ , and the universal derivation is the homomorphism $d$ defined by

ds=1\otimes s-s\otimes 1.

To see that this construction is equivalent to the previous one, note that $I$ is the kernel of the projection $S\otimes _{R}S\to S\otimes _{R}R$ given by $\Sigma s_{i}\otimes t_{i}\mapsto \Sigma s_{i}\cdot t_{i}\otimes 1$ . Thus we have:

S\otimes _{R}S\equiv I\oplus S\otimes _{R}R.

Then $S\otimes _{R}S/S\otimes _{R}R$ may be identified with $I$ by the map induced by the complementary projection $\Sigma s_{i}\otimes t_{i}\mapsto \Sigma s_{i}\otimes t_{i}-\Sigma s_{i}\cdot t_{i}\otimes 1$ . This identifies $I$ with the $S$ -module generated by the formal generators $ds$ for $s$ in $S$ , subject to $d$ being a homomorphism of $R$ -modules which sends each element of $R$ to zero. Taking the quotient by $I 2$ precisely imposes the Leibniz rule.

Examples and basic facts

For any commutative ring $R$ , the Kähler differentials of the polynomial ring $S=R[t_{1},\dots ,t_{n}]$ are a free $S$ -module of rank n generated by the differentials of the variables:

\Omega _{R[t_{1},\dots ,t_{n}]/R}^{1}=\bigoplus _{i=1}^{n}R[t_{1},\dots t_{n}]\,dt_{i}.

Kähler differentials are compatible with extension of scalars, in the sense that for a second $R$ -algebra $R'$ and for $S'=R'\otimes _{R}S$ , there is an isomorphism

\Omega _{S/R}\otimes _{S}S'\cong \Omega _{S'/R'}.

As a particular case of this, Kähler differentials are compatible with localizations, meaning that if $W$ is a multiplicative set in $S$ , then there is an isomorphism

W^{-1}\Omega _{S/R}\cong \Omega _{W^{-1}S/R}.

Given two ring homomorphisms $R\to S\to T$ , there is a short exact sequence of $T$ -modules

\Omega _{S/R}\otimes _{S}T\to \Omega _{T/R}\to \Omega _{T/S}\to 0.

If $T=S/I$ for some ideal $I$ , the term $\Omega _{T/S}$ vanishes and the sequence can be continued at the left as follows:

I/I^{2}{\xrightarrow {[f]\mapsto df\otimes 1}}\Omega _{S/R}\otimes _{S}T\to \Omega _{T/R}\to 0.

A generalization of these two short exact sequences is provided by the cotangent complex.

The latter sequence and the above computation for the polynomial ring allows the computation of the Kähler differentials of finitely generated $R$ -algebras $T=R[t_{1},\dots ,t_{n}]/(f_{1},\dots ,f_{m})$ . Briefly, these are generated by the differentials of the variables and have relations coming from the differentials of the equations. For example, for a single polynomial in a single variable,

\Omega _{(R[t]/(f))/R}\cong (R[t]\,dt\otimes R[t]/(f))/(df)\cong R[t]/(f,df)\,dt.

Kähler differentials for schemes

Because Kähler differentials are compatible with localization, they may be constructed on a general scheme by performing either of the two definitions above on affine open subschemes and gluing. However, the second definition has a geometric interpretation that globalizes immediately. In this interpretation, $I$ represents the ideal defining the diagonal in the fiber product of $Spec(S)$ with itself over $Spec(S) \to Spec(R)$ . This construction therefore has a more geometric flavor, in the sense that the notion of first infinitesimal neighbourhood of the diagonal is thereby captured, via functions vanishing modulo functions vanishing at least to second order (see cotangent space for related notions). Moreover, it extends to a general morphism of schemes $f \colon X \to Y$ by setting ${\mathcal {I}}$ to be the ideal of the diagonal in the fiber product $X\times _{Y}X$ . The cotangent sheaf $\Omega _{X/Y}={\mathcal {I}}/{\mathcal {I}}^{2}$ , together with the derivation $d\colon {\mathcal {O}}_{X}\to \Omega _{X/Y}$ defined analogously to before, is universal among $f^{-1}{\mathcal {O}}_{Y}$ -linear derivations of ${\mathcal {O}}_{X}$ -modules. If $U$ is an open affine subscheme of $X$ whose image in $Y$ is contained in an open affine subscheme $V$ , then the cotangent sheaf restricts to a sheaf on $U$ which is similarly universal. It is therefore the sheaf associated to the module of Kähler differentials for the rings underlying $U$ and $V$ .

Similar to the commutative algebra case, there exist exact sequences associated to morphisms of schemes. Given morphisms $f:X\to Y$ and $g:Y\to Z$ of schemes there is an exact sequence of sheaves on $X$

f^{*}\Omega _{Y/Z}\to \Omega _{X/Z}\to \Omega _{X/Y}\to 0

Also, if $Z\subset X$ is a closed subscheme given by the ideal sheaf ${\mathcal {I}}$ there is an exact sequence of sheaves

{\frac {\mathcal {I}}{{\mathcal {I}}^{2}}}\to \Omega _{X/Y}\otimes {\mathcal {O}}_{Z}\to \Omega _{Z/Y}\to 0

on $Z$

Examples

Cotangent Modules of a Projective Variety

Given a projective scheme $X\in {\text{Sch}}/\mathbb {k}$ , its cotangent sheaf can be computed from the sheafification of the cotangent module on the underlying graded algebra. For example, consider the complex curve

{\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(x^{n}+y^{n}-z^{n})}}\right)

then we can compute the cotangent module as

\Omega _{R/\mathbb {C} }={\frac {R\cdot dx\oplus R\cdot dy\oplus R\cdot dz}{nx^{n-1}dx+ny^{n-1}dy-nz^{n-1}dz}}

Then,

\Omega _{X/\mathbb {C} }={\widetilde {\Omega _{R/\mathbb {C} }}}

Morphisms of Schemes

Consider the morphism

X={\text{Spec}}\left({\frac {\mathbb {C} [t,x,y]}{(xy-t)}}\right)\to {\text{Spec}}(\mathbb {C} [t])=Y

in ${\text{Sch}}/\mathbb {C}$ . Then, using the first sequence we see that

{\widetilde {\mathbb {C} [t]}}\to {\widetilde {\frac {R\cdot dt\oplus R\cdot dx\oplus R\cdot dy}{ydx+xdy-dt}}}\to \Omega _{X/Y}\to 0

hence

\Omega _{X/Y}={\widetilde {\frac {R\cdot dx\oplus R\cdot dy}{ydx+xdy}}}

Higher differential forms and algebraic de Rham cohomology

de Rham complex

As before, fix a map $X\to Y$ . Differential forms of higher degree are defined as the exterior powers (over $\mathcal O_X$ ),

\Omega _{X/Y}^{n}:=\bigwedge ^{n}\Omega _{X/Y}.

The derivation ${\mathcal {O}}_{X}\to \Omega _{X/Y}$ extends in a natural way to a sequence of maps

0\to {\mathcal {O}}_{X}{\xrightarrow {d}}\Omega _{X/Y}^{1}{\xrightarrow {d}}\Omega _{X/Y}^{2}{\xrightarrow {d}}\cdots

satisfying $d\circ d=0$ . This is a cochain complex known as the de Rham complex.

The de Rham complex enjoys an additional multiplicative structure, the wedge product

\Omega _{X/Y}^{n}\otimes \Omega _{X/Y}^{m}\to \Omega _{X/Y}^{n+m}.

This turns the de Rham complex into a commutative differential graded algebra. It also has a coalgebra structure inherited from the one on the exterior algebra.^[2]

de Rham cohomology

The hypercohomology of the de Rham complex of sheaves is called the algebraic de Rham cohomology of $X$ over $Y$ and is denoted by

H_{\text{dR}}^{n}(X/Y)

or just $H_{\text{dR}}^{n}(X)$ if $Y$ is clear from the context. (In many situations, $Y$ is the spectrum of a field of characteristic zero.) Algebraic de Rham cohomology was introduced by Grothendieck (1966). It is closely related to crystalline cohomology.

As is familiar from coherent cohomology of other quasi-coherent sheaves, the computation of de Rham cohomology is simplified when $X = Spec S$ and $Y = Spec R$ are affine schemes. In this case, because affine schemes have no higher cohomology, $H_{\text{dR}}^{n}(X/Y)$ can be computed as the cohomology of the complex of abelian groups

0\to S{\xrightarrow {d}}\Omega _{S/R}^{1}{\xrightarrow {d}}\Omega _{S/R}^{2}{\xrightarrow {d}}\cdots

which is, termwise, the global sections of the sheaves $\Omega _{X/Y}^{r}$ .

To take a very particular example, suppose that $X = Spec Q [x, x - 1]$ is the multiplicative group over $Q$ . Because this is an affine scheme, hypercohomology reduces to ordinary cohomology. The algebraic de Rham complex is

\mathbf {Q} [x,x^{-1}]{\xrightarrow {d}}\mathbf {Q} [x,x^{-1}]\,dx.

The differential $d$ obeys the usual rules of calculus, meaning $d(x^{n})=nx^{n-1}\,dx$ . The kernel and cokernel compute algebraic de Rham cohomology, so $H_{\text{dR}}^{0}(X)=\mathbf {Q}$ , $H_{\text{dR}}^{1}(X)=\mathbf {Q} \cdot (dx/x)$ , and all other algebraic de Rham cohomology groups are zero. By way of comparison, the algebraic de Rham cohomology groups of $Y = Spec F p [x, x - 1]$ , are much larger, namely, $H_{\text{dR}}^{0}(Y)=\bigoplus _{k\in \mathbf {Z} }\mathbf {F} _{p}\cdot x^{kp}$ and $H_{\text{dR}}^{1}(Y)=\bigoplus _{k\in \mathbf {Z} }\mathbf {F} _{p}\cdot x^{kp-1}\,dx$ . Since the betti numbers of these cohomology groups are not what is expected, crystalline cohomology was developed to remedy this issue; it defines a weil-cohomology theory over finite fields.

Grothendieck's comparison theorem

If $X$ is smooth over $Y=\operatorname {Spec} \mathbf {C}$ , there is a natural comparison map

\Omega _{X/\mathbf {C} }^{r}(X)\to \Omega _{X^{\text{an}}}^{r}(X^{\text{an}})

between the Kähler (i.e., algebraic) differential forms on $X$ and the smooth (i.e., $C^{\infty }$ ) differential forms on $X^{\text{an}}$ , the complex manifold associated to X. This map need not be an isomorphism. However, the induced map

H_{\text{dR}}^{n}(X/\mathbf {C} )\to H_{\text{dR}}^{n}(X^{\text{an}})

between algebraic and smooth de Rham cohomology is an isomorphism, as was first shown by Grothendieck (1966). A proof using the concept of a Weil cohomology was given by Cisinski & Déglise (2013).

Applications

Canonical divisor

If $X$ is a smooth variety over a field $k$ , then $\Omega _{X/k}$ is a vector bundle (i.e., a locally free $\mathcal O_X$ -module) of rank equal to the dimension of $X$ . This implies, in particular, that

\omega _{X/k}:=\bigwedge ^{\dim X}\Omega _{X/k}

is a line bundle or, equivalently, a divisor. It is referred to as the canonical divisor. The canonical divisor is, as it turns out, a dualizing complex and therefore appears in various important theorems in algebraic geometry such as Serre duality or Verdier duality.

Classification of algebraic curves

The geometric genus of a smooth algebraic variety $X$ of dimension $d$ over a field $k$ is defined as the dimension

g:=\dim H^{0}(X,\Omega _{X/k}^{d}).

For curves, this purely algebraic definition agrees with the topological definition (for $k = C$ ) as the "number of handles" of the Riemann surface associated to X. There is a rather sharp trichotomy of geometric and arithmetic properties depending on the genus of a curve, for $g$ being 0 (rational curves), 1 (elliptic curves), and greater than 1 (hyperbolic Riemann surfaces, including hyperelliptic curves), respectively.

Tangent bundle and Riemann–Roch theorem

The tangent bundle of a smooth variety $X$ is, by definition, the dual of the cotangent sheaf $\Omega _{X/k}$ . The Riemann–Roch theorem and its far-reaching generalization, the Grothendieck–Riemann–Roch theorem, contain as a crucial ingredient the Todd class of the tangent bundle.

Unramified and smooth morphisms

The sheaf of differentials is related to various algebro-geometric notions. A morphism $f:X\to Y$ of schemes is unramified if and only if $\Omega _{X/Y}$ is zero.^[3] A special case of this assertion is that for a field $k$ , $K:=k[t]/f$ is separable over $k$ iff $\Omega _{K/k}=0$ , which can also be read off the above computation.

A morphism $f$ of finite type is a smooth morphism if it is flat and if $\Omega _{X/Y}$ is a locally free $\mathcal O_X$ -module of appropriate rank. The computation of $\Omega _{R[t_{1},\ldots ,t_{n}]/R}$ above shows that the projection from affine space $\mathbb {A} _{R}^{n}\to Spec(R)$ is smooth.

Periods

Periods are, broadly speaking, integrals of certain, arithmetically defined differential forms.^[4] The simplest example of a period is $2\pi i$ , which arises as $\int _{S^{1}}{\frac {dz}{z}}=2\pi i.$

Algebraic de Rham cohomology is used to construct periods as follows: For an algebraic variety $X$ defined over $Q$ , the above-mentioned compatibility with base-change yields a natural isomorphism

H_{\text{dR}}^{n}(X/\mathbf {Q} )\otimes _{\mathbf {Q} }\mathbf {C} =H_{\text{dR}}^{n}(X\otimes _{\mathbf {Q} }\mathbf {C} /\mathbf {C} ).

On the other hand, the right hand cohomology group is isomorphic to de Rham cohomology of the complex manifold $X^{\text{an}}$ associated to $X$ , denoted here $H_{\text{dR}}^{n}(X^{\text{an}}).$ Yet another classical result, de Rham's theorem, asserts an isomorphism of the latter cohomology group with singular cohomology (or sheaf cohomology) with complex coefficients, $H^{n}(X^{\text{an}},\mathbf {C} )$ , which by the universal coefficient theorem is in its turn isomorphic to $H^{n}(X^{\text{an}},\mathbf {Q} )\otimes _{\mathbf {Q} }\mathbf {C}$ . Composing these isomorphisms yields two rational vector spaces which, after tensoring with C, become isomorphic. Choosing bases of these rational subspaces (also called lattices), the determinant of the base-change matrix is a complex number, well defined up to multiplication by a rational number. Such numbers are periods.

Algebraic number theory

In algebraic number theory, Kähler differentials may be used to study the ramification in an extension of algebraic number fields. If $L / K$ is a finite extension with rings of integers $O$ and $o$ respectively then the different ideal $δ L / K$ , which encodes the ramification data, is the annihilator of the $O$ -module $Ω O / o$ :^[5]

\delta _{L/K}=\{x\in O:x\,dy=0{\text{ for all }}y\in O\}.

Related notions

Hochschild homology is a homology theory for associative rings which turns out to be closely related to Kähler differentials. This is because of the Hoschild-Kostant-Rosenberg theorem which states that the Hochschild homology $HH_{\bullet }(R)$ of an algebra of a smooth variety is isomorphic to the de-Rham complex $\Omega _{R/k}^{\bullet }$ for $k$ a field of characteristic $0$ . There is a derived enhancement of this theorem which states that the Hochschild homology of a dga is isomorphic to the derived de-Rham complex.

The de Rham–Witt complex is, in very rough terms, an enhancement of the de Rham complex for the ring of Witt vectors.

References

↑ Hartshorne (1977, p. 172)
↑ Laurent-Gengoux, C.; Pichereau, A.; Vanhaecke, P. (2013). Poisson structures. §3.2.3: Springer. ISBN 978-3-642-31090-4.
↑ Milne, James, Etale cohomology, Proposition I.3.5 ; the map $f$ is supposed to be locally of finite type for this statement.
↑ André, Yves (2004). Une introduction aux motifs. Partie III: Société Mathématique de France.
↑ Neukirch (1999, p. 201)

Cisinski, D.-C.; Déglise, F. (2013). "Mixed Weil cohomologies". Adv. Math. arXiv:0712.3291 . doi:10.1016/j.aim.2011.10.021.
Grothendieck, Alexander (1966), "On the de Rham cohomology of algebraic varieties", Institut des Hautes Études Scientifiques. Publications Mathématiques, 29 (29): 95–103, ISSN 0073-8301, MR 0199194, doi:10.1007/BF02684807 (letter to Atiyah, Oct. 14 1963)
Grothendieck, A. (1966), Letter to J. Tate (PDF) .
Grothendieck, Alexander (1968), "Crystals and the de Rham cohomology of schemes", in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L.; et al., Dix Exposés sur la Cohomologie des Schémas (PDF), Advanced studies in pure mathematics, 3, Amsterdam: North-Holland, pp. 306–358, MR 0269663
Johnson, J. (1969). "Kähler differentials and differential algebra". Annals of Mathematics. 89: 92–98. Zbl 0179.34302. doi:10.2307/1970810.
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge University Press.
Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Rosenlicht, M. (1976). "On Liouville's theory of elementary functions". Pacific J. Math. 65: 485–492. Zbl 0318.12107. doi:10.2140/pjm.1976.65.485.
Fu, G.; et al. (2011). "Some remarks on Kähler differentials and ordinary differentials in nonlinear control systems". Systems and Control Letters. 60: 699–703. doi:10.1016/j.sysconle.2011.05.006.

External links

Notes on p-adic algebraic de-Rham cohomology - gives many computations over characteristic 0 as motivation
A thread devoted to the relation on algebraic and analytic differential forms
Differentials (Stacks project)

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